Orbits of automorphism group of trinomial hypersurfaces

TL;DR

This paper studies the automorphism group actions on trinomial hypersurfaces, revealing finite orbit counts for nonrigid cases and providing detailed descriptions for certain classes, advancing understanding of their symmetry structures.

Contribution

It offers a comprehensive analysis of automorphism group orbits on trinomial hypersurfaces, including classifications for nonrigid and specific subclasses, which was previously not well-understood.

Findings

Nonrigid trinomial varieties have finitely many orbits.

Descriptions of orbits for flexible trinomial hypersurfaces.

Orbit classifications for hypersurfaces with a single variable of power one.

Abstract

Trinomial hypersurfaces form a natural class of affine algebraic varieties closely connected with varieties admitting a torus action of complexity one. We investigate orbits of the automorphism group on these hypersurfaces. We prove that each nonrigid trinomial variety has finite number of orbits. We investigate singular orbits and this gives us description of all orbits for some classes of flexible trinomial hypersurfaces. Also we obtain a description of orbits for a class of hypersurfaces having a unique variable with power one in the equation.